Data structure: \(O = (W, A, Z, Y)\)
Underlying data generating process, \(P_{U,X}\)
## W A Z Y
## Min. :-4.13666 Min. :0.000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.67951 1st Qu.:0.631 1st Qu.:0.0000 1st Qu.:0.0000
## Median :-0.02214 Median :2.022 Median :0.0000 Median :1.0000
## Mean :-0.01306 Mean :2.135 Mean :0.4874 Mean :0.6059
## 3rd Qu.: 0.65589 3rd Qu.:3.430 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.82870 Max. :5.000 Max. :1.0000 Max. :1.0000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.526 2.967 2.817 4.272 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.860 2.186 2.268 3.513 5.000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4559 1.7711 1.9452 3.1529 5.0000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.202 1.546 2.640 5.000
## Initial lambda mean: 0.008063207
## The selected undersmoothed lambda mean: 0.0001128359
## The scaler picked: 0.014
Estimating ATE(0.5,0) with z = 0
[Same as Simulation #1, but with posibility violation as the variance
of U_A decreased]
Data structure: \(O = (W, A, Z,
Y)\)
Underlying data generating process, \(P_{U,X}\)
## W A Z Y
## Min. :-4.039777 Min. :0.000 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.687173 1st Qu.:1.217 1st Qu.:0.000 1st Qu.:0.0000
## Median :-0.015235 Median :1.973 Median :0.000 Median :1.0000
## Mean :-0.006659 Mean :1.993 Mean :0.499 Mean :0.6582
## 3rd Qu.: 0.659240 3rd Qu.:2.731 3rd Qu.:1.000 3rd Qu.:1.0000
## Max. : 4.256538 Max. :5.000 Max. :1.000 Max. :1.0000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 2.013 2.671 2.701 3.362 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.511 2.219 2.220 2.906 5.000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.060 1.746 1.770 2.436 5.000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.5197 1.1890 1.2479 1.8568 4.8720
## Initial lambda mean: 0.01013305
## The selected undersmoothed lambda mean: 0.0002045276
## The scaler picked: 0.02018421
Estimating ATE(0.5,0) with z = 0
Estimating ATE(0.5,0) with z = 0
[Same as Simulation #1, but with a 3-way interaction]
Data structure: \(O = (W, A, Z,
Y)\)
Underlying data generating process, \(P_{U,X}\)
## W A Z Y
## Min. :-4.039777 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.687173 1st Qu.:0.5447 1st Qu.:0.0000 1st Qu.:0.0000
## Median :-0.015235 Median :1.9332 Median :0.0000 Median :1.0000
## Mean :-0.006659 Mean :2.0698 Mean :0.4954 Mean :0.5825
## 3rd Qu.: 0.659240 3rd Qu.:3.3509 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 4.256538 Max. :5.0000 Max. :1.0000 Max. :1.0000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.304 2.587 2.591 3.951 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.8416 2.1806 2.2611 3.5800 5.0000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.3482 1.6997 1.9047 3.0925 5.0000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.129 1.473 2.480 5.000
## Initial lambda mean: 0.007436044
## The selected undersmoothed lambda mean: 0.0001229773
## The scaler picked: 0.016538
Estimating ATE(0.5,0) with z = 0
Estimating ATE(0.5,0) with z = 0
[Same as Simulation #1, but with a 3-way interaction and positivity violations]
Data structure: \(O = (W, A, Z, Y)\)
Underlying data generating process, \(P_{U,X}\)
## W A Z Y
## Min. :-4.039777 Min. :0.000 Min. :0.000 Min. :0.0000
## 1st Qu.:-0.687173 1st Qu.:1.217 1st Qu.:0.000 1st Qu.:0.0000
## Median :-0.015235 Median :1.973 Median :0.000 Median :1.0000
## Mean :-0.006659 Mean :1.993 Mean :0.499 Mean :0.6582
## 3rd Qu.: 0.659240 3rd Qu.:2.731 3rd Qu.:1.000 3rd Qu.:1.0000
## Max. : 4.256538 Max. :5.000 Max. :1.000 Max. :1.0000
## Summary of A given W < -1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 2.013 2.671 2.701 3.362 5.000
## Summary of A given -1 < W <= 0:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.511 2.219 2.220 2.906 5.000
## Summary of A given 0 < W <= 1:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.060 1.746 1.770 2.436 5.000
## Summary of A given 1 < W:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.5197 1.1890 1.2479 1.8568 4.8720
## Initial lambda mean: 0.009272508
## The selected undersmoothed lambda mean: 0.0001897926
## The scaler picked: 0.02046831
Estimating ATE(0.5,0) with z = 0
Estimating ATE(0.5,0) with z = 0